In
mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the
natural numbers are the ordinary whole numbers used for
countingCounting is the action of finding the number of elements of a finite set of objects. The traditional way of counting consists of continually increasing a counter by a unit for every element of the set, in some order, while marking those elements to avoid visiting the same element more than once,...
("there are 6 coins on the table") and
orderingIn set theory, a total order, linear order, simple order, or ordering is a binary relation on some set X. The relation is transitive, antisymmetric, and total...
("this is the 3rd largest city in the country"). These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively (see English numerals). A more recent notion is that of a
nominal numberNominal numbers are numerals used for identification only. The numerical value is irrelevant, and they do not indicate quantity, rank, or any other measurement.-Definition:...
, which is used only for naming.
Properties of the natural numbers related to divisibility, such as the distribution of
prime numberA prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
s, are studied in
number theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
. Problems concerning counting and ordering, such as
partitionIn number theory and combinatorics, a partition of a positive integer n, also called an integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only in the order of their summands are considered to be the same partition; if order matters then the sum becomes a...
enumerationIn mathematics and theoretical computer science, the broadest and most abstract definition of an enumeration of a set is an exact listing of all of its elements . The restrictions imposed on the type of list used depend on the branch of mathematics and the context in which one is working...
, are studied in
combinatoricsCombinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
.
There is no universal agreement about whether to include zero in the set of natural numbers: some define the natural numbers to be the
positive integerThe integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s } = { { }, ,
{{refimprove|date=October 2011}}
In
mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the
natural numbers are the ordinary whole numbers used for
countingCounting is the action of finding the number of elements of a finite set of objects. The traditional way of counting consists of continually increasing a counter by a unit for every element of the set, in some order, while marking those elements to avoid visiting the same element more than once,...
("there are 6 coins on the table") and
orderingIn set theory, a total order, linear order, simple order, or ordering is a binary relation on some set X. The relation is transitive, antisymmetric, and total...
("this is the 3rd largest city in the country"). These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively (see English numerals). A more recent notion is that of a
nominal numberNominal numbers are numerals used for identification only. The numerical value is irrelevant, and they do not indicate quantity, rank, or any other measurement.-Definition:...
, which is used only for naming.
Properties of the natural numbers related to divisibility, such as the distribution of
prime numberA prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
s, are studied in
number theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
. Problems concerning counting and ordering, such as
partitionIn number theory and combinatorics, a partition of a positive integer n, also called an integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only in the order of their summands are considered to be the same partition; if order matters then the sum becomes a...
enumerationIn mathematics and theoretical computer science, the broadest and most abstract definition of an enumeration of a set is an exact listing of all of its elements . The restrictions imposed on the type of list used depend on the branch of mathematics and the context in which one is working...
, are studied in
combinatoricsCombinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
.
There is no universal agreement about whether to include zero in the set of natural numbers: some define the natural numbers to be the
positive integerThe integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s {{nowrap|1=
{{refimprove|date=October 2011}}
In
mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the
natural numbers are the ordinary whole numbers used for
countingCounting is the action of finding the number of elements of a finite set of objects. The traditional way of counting consists of continually increasing a counter by a unit for every element of the set, in some order, while marking those elements to avoid visiting the same element more than once,...
("there are 6 coins on the table") and
orderingIn set theory, a total order, linear order, simple order, or ordering is a binary relation on some set X. The relation is transitive, antisymmetric, and total...
("this is the 3rd largest city in the country"). These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively (see English numerals). A more recent notion is that of a
nominal numberNominal numbers are numerals used for identification only. The numerical value is irrelevant, and they do not indicate quantity, rank, or any other measurement.-Definition:...
, which is used only for naming.
Properties of the natural numbers related to divisibility, such as the distribution of
prime numberA prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
s, are studied in
number theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
. Problems concerning counting and ordering, such as
partitionIn number theory and combinatorics, a partition of a positive integer n, also called an integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only in the order of their summands are considered to be the same partition; if order matters then the sum becomes a...
enumerationIn mathematics and theoretical computer science, the broadest and most abstract definition of an enumeration of a set is an exact listing of all of its elements . The restrictions imposed on the type of list used depend on the branch of mathematics and the context in which one is working...
, are studied in
combinatoricsCombinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
.
There is no universal agreement about whether to include zero in the set of natural numbers: some define the natural numbers to be the
positive integerThe integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s {{nowrap|1=
{{refimprove|date=October 2011}}
In
mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the
natural numbers are the ordinary whole numbers used for
countingCounting is the action of finding the number of elements of a finite set of objects. The traditional way of counting consists of continually increasing a counter by a unit for every element of the set, in some order, while marking those elements to avoid visiting the same element more than once,...
("there are 6 coins on the table") and
orderingIn set theory, a total order, linear order, simple order, or ordering is a binary relation on some set X. The relation is transitive, antisymmetric, and total...
("this is the 3rd largest city in the country"). These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively (see English numerals). A more recent notion is that of a
nominal numberNominal numbers are numerals used for identification only. The numerical value is irrelevant, and they do not indicate quantity, rank, or any other measurement.-Definition:...
, which is used only for naming.
Properties of the natural numbers related to divisibility, such as the distribution of
prime numberA prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
s, are studied in
number theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
. Problems concerning counting and ordering, such as
partitionIn number theory and combinatorics, a partition of a positive integer n, also called an integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only in the order of their summands are considered to be the same partition; if order matters then the sum becomes a...
enumerationIn mathematics and theoretical computer science, the broadest and most abstract definition of an enumeration of a set is an exact listing of all of its elements . The restrictions imposed on the type of list used depend on the branch of mathematics and the context in which one is working...
, are studied in
combinatoricsCombinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
.
There is no universal agreement about whether to include zero in the set of natural numbers: some define the natural numbers to be the
positive integerThe integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s {{nowrap|1={{{num|1}}, {{num|2}}, {{num|3}}, ...}}}, while for others the term designates the
non-negative integers {{nowrap|1={{{num|0}}, 1, 2, 3, ...}}}. The former definition is the traditional one, with the latter definition first appearing in the 19th century. Some authors use the term "natural number" to exclude zero and "
whole numberWhole number is a term with inconsistent definitions by different authors. All distinguish whole numbers from fractions and numbers with fractional parts.Whole numbers may refer to:*natural numbers in sense — the positive integers...
" to include it; others use "whole number" in a way that excludes zero, or in a way that includes both zero and the negative integers.
History of natural numbers and the status of zero
The natural numbers had their origins in the words used to count things, beginning with the number 1.
The first major advance in abstraction was the use of
numeralsA numeral system is a writing system for expressing numbers, that is a mathematical notation for representing numbers of a given set, using graphemes or symbols in a consistent manner....
to represent numbers. This allowed systems to be developed for recording large numbers. The ancient
EgyptiansThe History of Ancient Egypt spans the period from the early predynastic settlements of the northern Nile Valley to the Roman conquest in 30 BC...
developed a powerful system of numerals with distinct
hieroglyphsEgyptian hieroglyphs were a formal writing system used by the ancient Egyptians that combined logographic and alphabetic elements. Egyptians used cursive hieroglyphs for religious literature on papyrus and wood...
for 1, 10, and all the powers of 10 up to over one million. A stone carving from
KarnakThe Karnak Temple Complex—usually called Karnak—comprises a vast mix of decayed temples, chapels, pylons, and other buildings, notably the Great Temple of Amun and a massive structure begun by Pharaoh Ramses II . Sacred Lake is part of the site as well. It is located near Luxor, some...
, dating from around 1500 BC and now at the
LouvreThe Musée du Louvre – in English, the Louvre Museum or simply the Louvre – is one of the world's largest museums, the most visited art museum in the world and a historic monument. A central landmark of Paris, it is located on the Right Bank of the Seine in the 1st arrondissement...
in Paris, depicts 276 as 2 hundreds, 7 tens, and 6 ones; and similarly for the number 4,622. The
BabyloniaBabylonia was an ancient cultural region in central-southern Mesopotamia , with Babylon as its capital. Babylonia emerged as a major power when Hammurabi Babylonia was an ancient cultural region in central-southern Mesopotamia (present-day Iraq), with Babylon as its capital. Babylonia emerged as...
ns had a
place-valuePositional notation or place-value notation is a method of representing or encoding numbers. Positional notation is distinguished from other notations for its use of the same symbol for the different orders of magnitude...
system based essentially on the numerals for 1 and 10.
A much later advance was the development of the idea that
zero0 is both a numberand the numerical digit used to represent that number in numerals.It fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, 0 is used as a placeholder in place value systems...
can be considered as a number, with its own numeral. The use of a zero
digitA digit is a symbol used in combinations to represent numbers in positional numeral systems. The name "digit" comes from the fact that the 10 digits of the hands correspond to the 10 symbols of the common base 10 number system, i.e...
in place-value notation (within other numbers) dates back as early as 700 BC by the Babylonians, but they omitted such a digit when it would have been the last symbol in the number. The
OlmecThe Olmec were the first major Pre-Columbian civilization in Mexico. They lived in the tropical lowlands of south-central Mexico, in the modern-day states of Veracruz and Tabasco....
and
Maya civilizationThe Maya is a Mesoamerican civilization, noted for the only known fully developed written language of the pre-Columbian Americas, as well as for its art, architecture, and mathematical and astronomical systems. Initially established during the Pre-Classic period The Maya is a Mesoamerican...
s used zero as a separate number as early as the 1st century BC, but this usage did not spread beyond
MesoamericaMesoamerica is a region and culture area in the Americas, extending approximately from central Mexico to Belize, Guatemala, El Salvador, Honduras, Nicaragua, and Costa Rica, within which a number of pre-Columbian societies flourished before the Spanish colonization of the Americas in the 15th and...
. The use of a numeral zero in modern times originated with the
IndiaIndia , officially the Republic of India , is a country in South Asia. It is the seventh-largest country by geographical area, the second-most populous country with over 1.2 billion people, and the most populous democracy in the world...
n mathematician
BrahmaguptaBrahmagupta was an Indian mathematician and astronomer who wrote many important works on mathematics and astronomy. His best known work is the Brāhmasphuṭasiddhānta , written in 628 in Bhinmal...
in 628. However, zero had been used as a number in the medieval
computusComputus is the calculation of the date of Easter in the Christian calendar. The name has been used for this procedure since the early Middle Ages, as it was one of the most important computations of the age....
(the calculation of the date of
EasterEaster is the central feast in the Christian liturgical year. According to the Canonical gospels, Jesus rose from the dead on the third day after his crucifixion. His resurrection is celebrated on Easter Day or Easter Sunday...
), beginning with
Dionysius ExiguusDionysius Exiguus was a 6th-century monk born in Scythia Minor, modern Dobruja shared by Romania and Bulgaria. He was a member of the Scythian monks community concentrated in Tomis, the major city of Scythia Minor...
in 525, without being denoted by a numeral (standard
Roman numeralsThe numeral system of ancient Rome, or Roman numerals, uses combinations of letters from the Latin alphabet to signify values. The numbers 1 to 10 can be expressed in Roman numerals as:...
do not have a symbol for zero); instead
nulla or
nullae, genitive of
nullus, the Latin word for "none", was employed to denote a zero value.
The first systematic study of numbers as
abstractionAbstraction is a process by which higher concepts are derived from the usage and classification of literal concepts, first principles, or other methods....
s (that is, as abstract
entitiesAn entity is something that has a distinct, separate existence, although it need not be a material existence. In particular, abstractions and legal fictions are usually regarded as entities. In general, there is also no presumption that an entity is animate.An entity could be viewed as a set...
) is usually credited to the
GreekAncient Greece is a civilization belonging to a period of Greek history that lasted from the Archaic period of the 8th to 6th centuries BC to the end of antiquity. Immediately following this period was the beginning of the Early Middle Ages and the Byzantine era. Included in Ancient Greece is the...
philosophers
PythagorasPythagoras of Samos was an Ionian Greek philosopher, mathematician, and founder of the religious movement called Pythagoreanism. Most of the information about Pythagoras was written down centuries after he lived, so very little reliable information is known about him...
and
ArchimedesArchimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an...
. Note that many Greek mathematicians did not consider 1 to be "a number", so to them
2 was the smallest number.
Independent studies also occurred at around the same time in
IndiaIndia , officially the Republic of India , is a country in South Asia. It is the seventh-largest country by geographical area, the second-most populous country with over 1.2 billion people, and the most populous democracy in the world...
,
ChinaChinese civilization may refer to:* China for more general discussion of the country.* Chinese culture* Greater China, the transnational community of ethnic Chinese.* History of China* Sinosphere, the area historically affected by Chinese culture...
, and
MesoamericaMesoamerica is a region and culture area in the Americas, extending approximately from central Mexico to Belize, Guatemala, El Salvador, Honduras, Nicaragua, and Costa Rica, within which a number of pre-Columbian societies flourished before the Spanish colonization of the Americas in the 15th and...
.{{fact|date=October 2011}}
Several
set-theoretical definitions of natural numbersSeveral ways have been proposed to define the natural numbers using set theory.- The contemporary standard :In standard, Zermelo-Fraenkel set theory the natural numbers...
were developed in the 19th century. With these definitions it was convenient to include 0 (corresponding to the
empty setIn mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...
) as a natural number. Including 0 is now the common convention among
set theoristsSet theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
,
logicIn philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
ians, and
computer scientistsComputer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...
. Many other mathematicians also include 0, although some have kept the older tradition and take 1 to be the first natural number. Sometimes the set of natural numbers with 0 included is called the set of
whole numberWhole number is a term with inconsistent definitions by different authors. All distinguish whole numbers from fractions and numbers with fractional parts.Whole numbers may refer to:*natural numbers in sense — the positive integers...
s or
counting numbers. On the other hand, while
integer being Latin for
whole, the integers usually stand for the negative and positive whole numbers (and zero) altogether.
Notation
Mathematicians use
N or
(an N in
blackboard boldBlackboard bold is a typeface style that is often used for certain symbols in mathematical texts, in which certain lines of the symbol are doubled. The symbols usually denote number sets...
, displayed as {{unicode|ℕ}} in Unicode) to refer to the set of all natural numbers. This set is countably infinite: it is
infinite but
countableIn mathematics, a countable set is a set with the same cardinality as some subset of the set of natural numbers. A set that is not countable is called uncountable. The term was originated by Georg Cantor...
by definition. This is also expressed by saying that the
cardinal numberIn mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite...
of the set is aleph-null
.
To be unambiguous about whether zero is included or not, sometimes an index (or superscript) "0" is added in the former case, and a superscript "
" or subscript "
" is added in the latter case:
-
-
(Sometimes, an index or superscript "+" is added to signify "positive". However, this is often used for "nonnegative" in other cases, as
R+ =
[0,∞) and
Z+ = { 0, 1, 2,... }, at least in European literature. The notation "
", however, is standard for nonzero, or rather, invertible elements.)
Some authors who exclude zero from the naturals use the terms
natural numbers with zero,
whole numbers, or
counting numbers, denoted
W, for the set of nonnegative integers. Others use the notation
P for the positive integers if there is no danger of confusing this with the prime numbers.
Set theorists often denote the set of all natural numbers including zero by a lower-case Greek letter
omegaOmega is the 24th and last letter of the Greek alphabet. In the Greek numeric system, it has a value of 800. The word literally means "great O" , as opposed to omicron, which means "little O"...
: ω. This stems from the identification of an
ordinal numberIn set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...
with the set of ordinals that are smaller. One may observe that adopting the von Neumann definition of ordinals and defining cardinal numbers as minimal ordinals among those with same
cardinality, one gets
.
Algebraic properties
The addition (+) and multiplication (×) operations on natural numbers have several algebraic properties:
- Closure
In mathematics, a set is said to be closed under some operation if performance of that operation on members of the set always produces a unique member of the same set. For example, the real numbers are closed under subtraction, but the natural numbers are not: 3 and 8 are both natural numbers, but...
under addition and multiplication: for all natural numbers a and b, both a + b and a × b are natural numbers.
- Associativity
In mathematics, associativity is a property of some binary operations. It means that, within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not...
: for all natural numbers a, b, and c, a + (b + c) = (a + b) + c and a × (b × c) = (a × b) × c.
- Commutativity
In mathematics an operation is commutative if changing the order of the operands does not change the end result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it...
: for all natural numbers a and b, a + b = b + a and a × b = b × a.
- Existence of identity element
In mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...
s: for every natural number a, a + 0 = a and a × 1 = a.
- Distributivity
In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalizes the distributive law from elementary algebra.For example:...
of multiplication over addition for all natural numbers a, b, and c, a × (b + c) = (a × b) + (a × c)
- No zero divisor
In abstract algebra, a nonzero element a of a ring is a left zero divisor if there exists a nonzero b such that ab = 0. Similarly, a nonzero element a of a ring is a right zero divisor if there exists a nonzero c such that ca = 0. An element that is both a left and a right zero divisor is simply...
s: if a and b are natural numbers such that a × b = 0 then a = 0 or b = 0
Properties
One can recursively define an addition on the natural numbers by setting
a + 0 =
a and {{nowrap|
a +
S(
b)}} = {{nowrap|
S(
a +
b)}} for all
a,
b. Here
S should be read as "successor". This turns the natural numbers {{nowrap|(
N, +)}} into a commutative
monoidIn abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as they are naturally semigroups with identity. Monoids occur in several branches of mathematics; for...
with
identity elementIn mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...
0, the so-called
free monoidIn mathematics, the idea of a free object is one of the basic concepts of abstract algebra. It is a part of universal algebra, in the sense that it relates to all types of algebraic structure . It also has a formulation in terms of category theory, although this is in yet more abstract terms....
with one generator. This monoid satisfies the
cancellation propertyIn mathematics, the notion of cancellative is a generalization of the notion of invertible.An element a in a magma has the left cancellation property if for all b and c in M, a * b = a * c always implies b = c.An element a in a magma has the right cancellation...
and can be embedded in a
groupIn mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
. The smallest group containing the natural numbers is the
integerThe integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s.
If we define 1 :=
S(0), then
b + 1 =
b +
S(0) =
S(
b + 0) =
S(
b). That is,
b + 1 is simply the successor of
b.
Analogously, given that addition has been defined, a
multiplicationMultiplication is the mathematical operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic ....
× can be defined via
a × 0 = 0 and
a × S(
b) = (
a ×
b) +
a. This turns {{nowrap|(
N*, ×)}} into a free commutative monoid with identity element 1; a generator set for this monoid is the set of
prime numberA prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
s. Addition and multiplication are compatible, which is expressed in the
distribution lawIn mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalizes the distributive law from elementary algebra.For example:...
:
{{nowrap|
a × (
b +
c)}} = {{nowrap|(
a ×
b) + (
a ×
c)}}. These properties of addition and multiplication make the natural numbers an instance of a commutative
semiringIn abstract algebra, a semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse...
. Semirings are an algebraic generalization of the natural numbers where multiplication is not necessarily commutative. The lack of additive inverses, which is equivalent to the fact that
N is not closed under subtraction, means that
N is
not a
ringIn mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
; instead it is a
semiringIn abstract algebra, a semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse...
(also known as a
rig).
If we interpret the natural numbers as "excluding 0", and "starting at 1", the definitions of + and × are as above, except that we start with
a + 1 =
S(
a) and {{nowrap|
a × 1 {{=}}
a}}.
For the remainder of the article, we write
ab to indicate the product
a ×
b, and we also assume the standard
order of operationsIn mathematics and computer programming, the order of operations is a rule used to clarify unambiguously which procedures should be performed first in a given mathematical expression....
.
Furthermore, one defines a
total orderIn set theory, a total order, linear order, simple order, or ordering is a binary relation on some set X. The relation is transitive, antisymmetric, and total...
on the natural numbers by writing
a ≤
b if and only if there exists another natural number
c with
a +
c =
b. This order is compatible with the arithmetical operations in the following sense: if
a,
b and
c are natural numbers and
a ≤
b, then {{nowrap|
a +
c}} ≤ {{nowrap|
b +
c}} and {{nowrap|
ac ≤
bc}}. An important property of the natural numbers is that they are
well-orderIn mathematics, a well-order relation on a set S is a strict total order on S with the property that every non-empty subset of S has a least element in this ordering. Equivalently, a well-ordering is a well-founded strict total order...
ed: every non-empty set of natural numbers has a least element. The rank among well-ordered sets is expressed by an
ordinal numberIn set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...
; for the natural numbers this is expressed as "ω".
While it is in general not possible to divide one natural number by another and get a natural number as result, the procedure of
divisionright|thumb|200px|20 \div 4=5In mathematics, especially in elementary arithmetic, division is an arithmetic operation.Specifically, if c times b equals a, written:c \times b = a\,...
with remainder is available as a substitute: for any two natural numbers
a and
b with
b ≠ 0 we can find natural numbers
q and
r such that
- a = bq + r and r < b.
The number
q is called the
quotientIn mathematics, a quotient is the result of division. For example, when dividing 6 by 3, the quotient is 2, while 6 is called the dividend, and 3 the divisor. The quotient further is expressed as the number of times the divisor divides into the dividend e.g. The quotient of 6 and 2 is also 3.A...
and
r is called the
remainderIn arithmetic, the remainder is the amount "left over" after the division of two integers which cannot be expressed with an integer quotient....
of division of
a by
b. The numbers
q and
r are uniquely determined by
a and
b. This, the
Division algorithmIn mathematics, and more particularly in arithmetic, the usual process of division of integers producing a quotient and a remainder can be specified precisely by a theorem stating that these exist uniquely with given properties. An integer division algorithm is any effective method for producing...
, is key to several other properties (divisibility), algorithms (such as the
Euclidean algorithmIn mathematics, the Euclidean algorithm is an efficient method for computing the greatest common divisor of two integers, also known as the greatest common factor or highest common factor...
), and ideas in number theory.
Generalizations
Two generalizations of natural numbers arise from the two uses:
- A natural number can be used to express the size of a finite set; more generally a cardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite...
is a measure for the size of a set also suitable for infinite sets; this refers to a concept of "size" such that if there is a bijection between two sets they have the same sizeIn mathematics, two sets are equinumerous if they have the same cardinality, i.e., if there exists a bijection f : A → B for sets A and B. This is usually denoted A \approx B \, or A \sim B....
. The set of natural numbers itself and any other countably infinite set has cardinality aleph-null (
).
- Linguistic ordinal numbers "first", "second", "third" can be assigned to the elements of a totally ordered finite set, and also to the elements of well-ordered countably infinite sets like the set of natural numbers itself. This can be generalized to ordinal number
In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...
s which describe the position of an element in a well-ordered set in general. An ordinal number is also used to describe the "size" of a well-ordered set, in a sense different from cardinality: if there is an order isomorphismIn the mathematical field of order theory an order isomorphism is a special kind of monotone function that constitutes a suitable notion of isomorphism for partially ordered sets . Whenever two posets are order isomorphic, they can be considered to be "essentially the same" in the sense that one of...
between two well-ordered sets they have the same ordinal number. The first ordinal number that is not a natural number is expressed as 
; this is also the ordinal number of the set of natural numbers itself.
Many well-ordered sets with cardinal number
have an ordinal number greater than ω (the latter is the lowest possible). The least ordinal of cardinality
(i.e., the
initial ordinalThe von Neumann cardinal assignment is a cardinal assignment which uses ordinal numbers. For a well-ordered set U, we define its cardinal number to be the smallest ordinal number equinumerous to U. More precisely:...
) is
.
For
finite well-ordered sets, there is one-to-one correspondence between ordinal and cardinal numbers; therefore they can both be expressed by the same natural number, the number of elements of the set. This number can also be used to describe the position of an element in a larger finite, or an infinite,
sequenceIn mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
.
Hypernatural numbers are part of a non-standard model of arithmetic due to Skolem.
Other generalizations are discussed in the article on
numberA number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....
s.
Formal definitions
{{Main|Set-theoretic definition of natural numbers}}
Historically, the precise mathematical definition of the natural numbers developed with some difficulty. The Peano axioms state conditions that any successful definition must satisfy. Certain constructions show that, given
set theorySet theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
,
modelsIn mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
of the Peano postulates must exist.
Peano axioms
{{Main|Peano axioms}}
The
Peano axiomsIn mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are a set of axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano...
give a formal theory of the natural numbers. The axioms are:
- There is a natural number 0.
- Every natural number a has a natural number successor, denoted by S(a). Intuitively, S(a) is a+1.
- There is no natural number whose successor is 0.
- S is injective, i.e. distinct natural numbers have distinct successors: if a ≠ b, then S(a) ≠ S(b).
- If a property is possessed by 0 and also by the successor of every natural number which possesses it, then it is possessed by all natural numbers. (This postulate ensures that the proof technique of mathematical induction
Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers...
is valid.)
It should be noted that the "0" in the above definition need not correspond to what we normally consider to be the number zero. "0" simply means some object that when combined with an appropriate successor function, satisfies the Peano axioms. All systems that satisfy these axioms are isomorphic, the name "0" is used here for the first element (the term "zeroth element" has been suggested to leave "first element" to "1", "second element" to "2", etc.), which is the only element that is not a successor. For example, the natural numbers starting with one also satisfy the axioms, if the symbol 0 is interpreted as the natural number 1, the symbol
S(
0) as the number 2, etc. In fact, in Peano's original formulation, the first natural number
was 1.
A standard construction
A standard construction in
set theorySet theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
, a special case of the von Neumann ordinal construction, is to define the natural numbers as follows:
- We set 0 := { }, the empty set
In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...
,
- and define S(a) = a ∪ {a} for every set a. S(a) is the successor of a, and S is called the successor function.
- By the axiom of infinity
In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of infinity is one of the axioms of Zermelo-Fraenkel set theory...
, the set of all natural numbers exists and is the intersection of all sets containing 0 which are closed under this successor function. This then satisfies the Peano axiomsIn mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are a set of axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano...
.
- Each natural number is then equal to the set of all natural numbers less than it, so that
- 0 = { }
- 1 = {0} = {{ }}
- 2 = {0, 1} = {0, {0}} = { { }, {{ }} }
- 3 = {0, 1, 2} = {0, {0}, {0, {0}}} = { { }, {{ }},
{{refimprove|date=October 2011}}
In mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the natural numbers are the ordinary whole numbers used for countingCounting is the action of finding the number of elements of a finite set of objects. The traditional way of counting consists of continually increasing a counter by a unit for every element of the set, in some order, while marking those elements to avoid visiting the same element more than once,...
("there are 6 coins on the table") and orderingIn set theory, a total order, linear order, simple order, or ordering is a binary relation on some set X. The relation is transitive, antisymmetric, and total...
("this is the 3rd largest city in the country"). These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively (see English numerals). A more recent notion is that of a nominal numberNominal numbers are numerals used for identification only. The numerical value is irrelevant, and they do not indicate quantity, rank, or any other measurement.-Definition:...
, which is used only for naming.
Properties of the natural numbers related to divisibility, such as the distribution of prime numberA prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
s, are studied in number theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
. Problems concerning counting and ordering, such as partitionIn number theory and combinatorics, a partition of a positive integer n, also called an integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only in the order of their summands are considered to be the same partition; if order matters then the sum becomes a...
enumerationIn mathematics and theoretical computer science, the broadest and most abstract definition of an enumeration of a set is an exact listing of all of its elements . The restrictions imposed on the type of list used depend on the branch of mathematics and the context in which one is working...
, are studied in combinatoricsCombinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
.
There is no universal agreement about whether to include zero in the set of natural numbers: some define the natural numbers to be the positive integerThe integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s {{nowrap|1=
{{refimprove|date=October 2011}}
In mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the natural numbers are the ordinary whole numbers used for countingCounting is the action of finding the number of elements of a finite set of objects. The traditional way of counting consists of continually increasing a counter by a unit for every element of the set, in some order, while marking those elements to avoid visiting the same element more than once,...
("there are 6 coins on the table") and orderingIn set theory, a total order, linear order, simple order, or ordering is a binary relation on some set X. The relation is transitive, antisymmetric, and total...
("this is the 3rd largest city in the country"). These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively (see English numerals). A more recent notion is that of a nominal numberNominal numbers are numerals used for identification only. The numerical value is irrelevant, and they do not indicate quantity, rank, or any other measurement.-Definition:...
, which is used only for naming.
Properties of the natural numbers related to divisibility, such as the distribution of prime numberA prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
s, are studied in number theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
. Problems concerning counting and ordering, such as partitionIn number theory and combinatorics, a partition of a positive integer n, also called an integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only in the order of their summands are considered to be the same partition; if order matters then the sum becomes a...
enumerationIn mathematics and theoretical computer science, the broadest and most abstract definition of an enumeration of a set is an exact listing of all of its elements . The restrictions imposed on the type of list used depend on the branch of mathematics and the context in which one is working...
, are studied in combinatoricsCombinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
.
There is no universal agreement about whether to include zero in the set of natural numbers: some define the natural numbers to be the positive integerThe integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s {{nowrap|1=
{{refimprove|date=October 2011}}
In mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the natural numbers are the ordinary whole numbers used for countingCounting is the action of finding the number of elements of a finite set of objects. The traditional way of counting consists of continually increasing a counter by a unit for every element of the set, in some order, while marking those elements to avoid visiting the same element more than once,...
("there are 6 coins on the table") and orderingIn set theory, a total order, linear order, simple order, or ordering is a binary relation on some set X. The relation is transitive, antisymmetric, and total...
("this is the 3rd largest city in the country"). These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively (see English numerals). A more recent notion is that of a nominal numberNominal numbers are numerals used for identification only. The numerical value is irrelevant, and they do not indicate quantity, rank, or any other measurement.-Definition:...
, which is used only for naming.
Properties of the natural numbers related to divisibility, such as the distribution of prime numberA prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
s, are studied in number theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
. Problems concerning counting and ordering, such as partitionIn number theory and combinatorics, a partition of a positive integer n, also called an integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only in the order of their summands are considered to be the same partition; if order matters then the sum becomes a...
enumerationIn mathematics and theoretical computer science, the broadest and most abstract definition of an enumeration of a set is an exact listing of all of its elements . The restrictions imposed on the type of list used depend on the branch of mathematics and the context in which one is working...
, are studied in combinatoricsCombinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
.
There is no universal agreement about whether to include zero in the set of natural numbers: some define the natural numbers to be the positive integerThe integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s {{nowrap|1={{{num|1}}, {{num|2}}, {{num|3}}, ...}}}, while for others the term designates the non-negative integers {{nowrap|1={{{num|0}}, 1, 2, 3, ...}}}. The former definition is the traditional one, with the latter definition first appearing in the 19th century. Some authors use the term "natural number" to exclude zero and "whole numberWhole number is a term with inconsistent definitions by different authors. All distinguish whole numbers from fractions and numbers with fractional parts.Whole numbers may refer to:*natural numbers in sense — the positive integers...
" to include it; others use "whole number" in a way that excludes zero, or in a way that includes both zero and the negative integers.
History of natural numbers and the status of zero
The natural numbers had their origins in the words used to count things, beginning with the number 1.
The first major advance in abstraction was the use of numeralsA numeral system is a writing system for expressing numbers, that is a mathematical notation for representing numbers of a given set, using graphemes or symbols in a consistent manner....
to represent numbers. This allowed systems to be developed for recording large numbers. The ancient EgyptiansThe History of Ancient Egypt spans the period from the early predynastic settlements of the northern Nile Valley to the Roman conquest in 30 BC...
developed a powerful system of numerals with distinct hieroglyphsEgyptian hieroglyphs were a formal writing system used by the ancient Egyptians that combined logographic and alphabetic elements. Egyptians used cursive hieroglyphs for religious literature on papyrus and wood...
for 1, 10, and all the powers of 10 up to over one million. A stone carving from KarnakThe Karnak Temple Complex—usually called Karnak—comprises a vast mix of decayed temples, chapels, pylons, and other buildings, notably the Great Temple of Amun and a massive structure begun by Pharaoh Ramses II . Sacred Lake is part of the site as well. It is located near Luxor, some...
, dating from around 1500 BC and now at the LouvreThe Musée du Louvre – in English, the Louvre Museum or simply the Louvre – is one of the world's largest museums, the most visited art museum in the world and a historic monument. A central landmark of Paris, it is located on the Right Bank of the Seine in the 1st arrondissement...
in Paris, depicts 276 as 2 hundreds, 7 tens, and 6 ones; and similarly for the number 4,622. The BabyloniaBabylonia was an ancient cultural region in central-southern Mesopotamia , with Babylon as its capital. Babylonia emerged as a major power when Hammurabi Babylonia was an ancient cultural region in central-southern Mesopotamia (present-day Iraq), with Babylon as its capital. Babylonia emerged as...
ns had a place-valuePositional notation or place-value notation is a method of representing or encoding numbers. Positional notation is distinguished from other notations for its use of the same symbol for the different orders of magnitude...
system based essentially on the numerals for 1 and 10.
A much later advance was the development of the idea that zero0 is both a numberand the numerical digit used to represent that number in numerals.It fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, 0 is used as a placeholder in place value systems...
can be considered as a number, with its own numeral. The use of a zero digitA digit is a symbol used in combinations to represent numbers in positional numeral systems. The name "digit" comes from the fact that the 10 digits of the hands correspond to the 10 symbols of the common base 10 number system, i.e...
in place-value notation (within other numbers) dates back as early as 700 BC by the Babylonians, but they omitted such a digit when it would have been the last symbol in the number. The OlmecThe Olmec were the first major Pre-Columbian civilization in Mexico. They lived in the tropical lowlands of south-central Mexico, in the modern-day states of Veracruz and Tabasco....
and Maya civilizationThe Maya is a Mesoamerican civilization, noted for the only known fully developed written language of the pre-Columbian Americas, as well as for its art, architecture, and mathematical and astronomical systems. Initially established during the Pre-Classic period The Maya is a Mesoamerican...
s used zero as a separate number as early as the 1st century BC, but this usage did not spread beyond MesoamericaMesoamerica is a region and culture area in the Americas, extending approximately from central Mexico to Belize, Guatemala, El Salvador, Honduras, Nicaragua, and Costa Rica, within which a number of pre-Columbian societies flourished before the Spanish colonization of the Americas in the 15th and...
. The use of a numeral zero in modern times originated with the IndiaIndia , officially the Republic of India , is a country in South Asia. It is the seventh-largest country by geographical area, the second-most populous country with over 1.2 billion people, and the most populous democracy in the world...
n mathematician BrahmaguptaBrahmagupta was an Indian mathematician and astronomer who wrote many important works on mathematics and astronomy. His best known work is the Brāhmasphuṭasiddhānta , written in 628 in Bhinmal...
in 628. However, zero had been used as a number in the medieval computusComputus is the calculation of the date of Easter in the Christian calendar. The name has been used for this procedure since the early Middle Ages, as it was one of the most important computations of the age....
(the calculation of the date of EasterEaster is the central feast in the Christian liturgical year. According to the Canonical gospels, Jesus rose from the dead on the third day after his crucifixion. His resurrection is celebrated on Easter Day or Easter Sunday...
), beginning with Dionysius ExiguusDionysius Exiguus was a 6th-century monk born in Scythia Minor, modern Dobruja shared by Romania and Bulgaria. He was a member of the Scythian monks community concentrated in Tomis, the major city of Scythia Minor...
in 525, without being denoted by a numeral (standard Roman numeralsThe numeral system of ancient Rome, or Roman numerals, uses combinations of letters from the Latin alphabet to signify values. The numbers 1 to 10 can be expressed in Roman numerals as:...
do not have a symbol for zero); instead nulla or nullae, genitive of nullus, the Latin word for "none", was employed to denote a zero value.
The first systematic study of numbers as abstractionAbstraction is a process by which higher concepts are derived from the usage and classification of literal concepts, first principles, or other methods....
s (that is, as abstract entitiesAn entity is something that has a distinct, separate existence, although it need not be a material existence. In particular, abstractions and legal fictions are usually regarded as entities. In general, there is also no presumption that an entity is animate.An entity could be viewed as a set...
) is usually credited to the GreekAncient Greece is a civilization belonging to a period of Greek history that lasted from the Archaic period of the 8th to 6th centuries BC to the end of antiquity. Immediately following this period was the beginning of the Early Middle Ages and the Byzantine era. Included in Ancient Greece is the...
philosophers PythagorasPythagoras of Samos was an Ionian Greek philosopher, mathematician, and founder of the religious movement called Pythagoreanism. Most of the information about Pythagoras was written down centuries after he lived, so very little reliable information is known about him...
and ArchimedesArchimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an...
. Note that many Greek mathematicians did not consider 1 to be "a number", so to them 2 was the smallest number.
Independent studies also occurred at around the same time in IndiaIndia , officially the Republic of India , is a country in South Asia. It is the seventh-largest country by geographical area, the second-most populous country with over 1.2 billion people, and the most populous democracy in the world...
, ChinaChinese civilization may refer to:* China for more general discussion of the country.* Chinese culture* Greater China, the transnational community of ethnic Chinese.* History of China* Sinosphere, the area historically affected by Chinese culture...
, and MesoamericaMesoamerica is a region and culture area in the Americas, extending approximately from central Mexico to Belize, Guatemala, El Salvador, Honduras, Nicaragua, and Costa Rica, within which a number of pre-Columbian societies flourished before the Spanish colonization of the Americas in the 15th and...
.{{fact|date=October 2011}}
Several set-theoretical definitions of natural numbersSeveral ways have been proposed to define the natural numbers using set theory.- The contemporary standard :In standard, Zermelo-Fraenkel set theory the natural numbers...
were developed in the 19th century. With these definitions it was convenient to include 0 (corresponding to the empty setIn mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...
) as a natural number. Including 0 is now the common convention among set theoristsSet theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
, logicIn philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
ians, and computer scientistsComputer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...
. Many other mathematicians also include 0, although some have kept the older tradition and take 1 to be the first natural number. Sometimes the set of natural numbers with 0 included is called the set of whole numberWhole number is a term with inconsistent definitions by different authors. All distinguish whole numbers from fractions and numbers with fractional parts.Whole numbers may refer to:*natural numbers in sense — the positive integers...
s or counting numbers. On the other hand, while integer being Latin for whole, the integers usually stand for the negative and positive whole numbers (and zero) altogether.
Notation
Mathematicians use N or (an N in blackboard boldBlackboard bold is a typeface style that is often used for certain symbols in mathematical texts, in which certain lines of the symbol are doubled. The symbols usually denote number sets...
, displayed as {{unicode|ℕ}} in Unicode) to refer to the set of all natural numbers. This set is countably infinite: it is infinite but countableIn mathematics, a countable set is a set with the same cardinality as some subset of the set of natural numbers. A set that is not countable is called uncountable. The term was originated by Georg Cantor...
by definition. This is also expressed by saying that the cardinal numberIn mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite...
of the set is aleph-null .
To be unambiguous about whether zero is included or not, sometimes an index (or superscript) "0" is added in the former case, and a superscript "" or subscript "" is added in the latter case:
-
-
(Sometimes, an index or superscript "+" is added to signify "positive". However, this is often used for "nonnegative" in other cases, as R+ = [0,∞) and Z+ = { 0, 1, 2,... }, at least in European literature. The notation "", however, is standard for nonzero, or rather, invertible elements.)
Some authors who exclude zero from the naturals use the terms natural numbers with zero, whole numbers, or counting numbers, denoted W, for the set of nonnegative integers. Others use the notation P for the positive integers if there is no danger of confusing this with the prime numbers.
Set theorists often denote the set of all natural numbers including zero by a lower-case Greek letter omegaOmega is the 24th and last letter of the Greek alphabet. In the Greek numeric system, it has a value of 800. The word literally means "great O" , as opposed to omicron, which means "little O"...
: ω. This stems from the identification of an ordinal numberIn set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...
with the set of ordinals that are smaller. One may observe that adopting the von Neumann definition of ordinals and defining cardinal numbers as minimal ordinals among those with same cardinality, one gets .
Algebraic properties
The addition (+) and multiplication (×) operations on natural numbers have several algebraic properties:
- Closure
In mathematics, a set is said to be closed under some operation if performance of that operation on members of the set always produces a unique member of the same set. For example, the real numbers are closed under subtraction, but the natural numbers are not: 3 and 8 are both natural numbers, but...
under addition and multiplication: for all natural numbers a and b, both a + b and a × b are natural numbers.
- Associativity
In mathematics, associativity is a property of some binary operations. It means that, within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not...
: for all natural numbers a, b, and c, a + (b + c) = (a + b) + c and a × (b × c) = (a × b) × c.
- Commutativity
In mathematics an operation is commutative if changing the order of the operands does not change the end result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it...
: for all natural numbers a and b, a + b = b + a and a × b = b × a.
- Existence of identity element
In mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...
s: for every natural number a, a + 0 = a and a × 1 = a.
- Distributivity
In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalizes the distributive law from elementary algebra.For example:...
of multiplication over addition for all natural numbers a, b, and c, a × (b + c) = (a × b) + (a × c)
- No zero divisor
In abstract algebra, a nonzero element a of a ring is a left zero divisor if there exists a nonzero b such that ab = 0. Similarly, a nonzero element a of a ring is a right zero divisor if there exists a nonzero c such that ca = 0. An element that is both a left and a right zero divisor is simply...
s: if a and b are natural numbers such that a × b = 0 then a = 0 or b = 0
Properties
One can recursively define an addition on the natural numbers by setting a + 0 = a and {{nowrap|a + S(b)}} = {{nowrap|S(a + b)}} for all a, b. Here S should be read as "successor". This turns the natural numbers {{nowrap|(N, +)}} into a commutative monoidIn abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as they are naturally semigroups with identity. Monoids occur in several branches of mathematics; for...
with identity elementIn mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...
0, the so-called free monoidIn mathematics, the idea of a free object is one of the basic concepts of abstract algebra. It is a part of universal algebra, in the sense that it relates to all types of algebraic structure . It also has a formulation in terms of category theory, although this is in yet more abstract terms....
with one generator. This monoid satisfies the cancellation propertyIn mathematics, the notion of cancellative is a generalization of the notion of invertible.An element a in a magma has the left cancellation property if for all b and c in M, a * b = a * c always implies b = c.An element a in a magma has the right cancellation...
and can be embedded in a groupIn mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
. The smallest group containing the natural numbers is the integerThe integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s.
If we define 1 := S(0), then b + 1 = b + S(0) = S(b + 0) = S(b). That is, b + 1 is simply the successor of b.
Analogously, given that addition has been defined, a multiplicationMultiplication is the mathematical operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic ....
× can be defined via a × 0 = 0 and a × S(b) = (a × b) + a. This turns {{nowrap|(N*, ×)}} into a free commutative monoid with identity element 1; a generator set for this monoid is the set of prime numberA prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
s. Addition and multiplication are compatible, which is expressed in the distribution lawIn mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalizes the distributive law from elementary algebra.For example:...
:
{{nowrap|a × (b + c)}} = {{nowrap|(a × b) + (a × c)}}. These properties of addition and multiplication make the natural numbers an instance of a commutative semiringIn abstract algebra, a semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse...
. Semirings are an algebraic generalization of the natural numbers where multiplication is not necessarily commutative. The lack of additive inverses, which is equivalent to the fact that N is not closed under subtraction, means that N is not a ringIn mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
; instead it is a semiringIn abstract algebra, a semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse...
(also known as a rig).
If we interpret the natural numbers as "excluding 0", and "starting at 1", the definitions of + and × are as above, except that we start with a + 1 = S(a) and {{nowrap|a × 1 {{=}} a}}.
For the remainder of the article, we write ab to indicate the product a × b, and we also assume the standard order of operationsIn mathematics and computer programming, the order of operations is a rule used to clarify unambiguously which procedures should be performed first in a given mathematical expression....
.
Furthermore, one defines a total orderIn set theory, a total order, linear order, simple order, or ordering is a binary relation on some set X. The relation is transitive, antisymmetric, and total...
on the natural numbers by writing a ≤ b if and only if there exists another natural number c with a + c = b. This order is compatible with the arithmetical operations in the following sense: if a, b and c are natural numbers and a ≤ b, then {{nowrap|a + c}} ≤ {{nowrap|b + c}} and {{nowrap|ac ≤ bc}}. An important property of the natural numbers is that they are well-orderIn mathematics, a well-order relation on a set S is a strict total order on S with the property that every non-empty subset of S has a least element in this ordering. Equivalently, a well-ordering is a well-founded strict total order...
ed: every non-empty set of natural numbers has a least element. The rank among well-ordered sets is expressed by an ordinal numberIn set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...
; for the natural numbers this is expressed as "ω".
While it is in general not possible to divide one natural number by another and get a natural number as result, the procedure of divisionright|thumb|200px|20 \div 4=5In mathematics, especially in elementary arithmetic, division is an arithmetic operation.Specifically, if c times b equals a, written:c \times b = a\,...
with remainder is available as a substitute: for any two natural numbers a and b with b ≠ 0 we can find natural numbers q and r such that
- a = bq + r and r < b.
The number q is called the quotientIn mathematics, a quotient is the result of division. For example, when dividing 6 by 3, the quotient is 2, while 6 is called the dividend, and 3 the divisor. The quotient further is expressed as the number of times the divisor divides into the dividend e.g. The quotient of 6 and 2 is also 3.A...
and r is called the remainderIn arithmetic, the remainder is the amount "left over" after the division of two integers which cannot be expressed with an integer quotient....
of division of a by b. The numbers q and r are uniquely determined by a and b. This, the Division algorithmIn mathematics, and more particularly in arithmetic, the usual process of division of integers producing a quotient and a remainder can be specified precisely by a theorem stating that these exist uniquely with given properties. An integer division algorithm is any effective method for producing...
, is key to several other properties (divisibility), algorithms (such as the Euclidean algorithmIn mathematics, the Euclidean algorithm is an efficient method for computing the greatest common divisor of two integers, also known as the greatest common factor or highest common factor...
), and ideas in number theory.
Generalizations
Two generalizations of natural numbers arise from the two uses:
- A natural number can be used to express the size of a finite set; more generally a cardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite...
is a measure for the size of a set also suitable for infinite sets; this refers to a concept of "size" such that if there is a bijection between two sets they have the same sizeIn mathematics, two sets are equinumerous if they have the same cardinality, i.e., if there exists a bijection f : A → B for sets A and B. This is usually denoted A \approx B \, or A \sim B....
. The set of natural numbers itself and any other countably infinite set has cardinality aleph-null ().
- Linguistic ordinal numbers "first", "second", "third" can be assigned to the elements of a totally ordered finite set, and also to the elements of well-ordered countably infinite sets like the set of natural numbers itself. This can be generalized to ordinal number
In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...
s which describe the position of an element in a well-ordered set in general. An ordinal number is also used to describe the "size" of a well-ordered set, in a sense different from cardinality: if there is an order isomorphismIn the mathematical field of order theory an order isomorphism is a special kind of monotone function that constitutes a suitable notion of isomorphism for partially ordered sets . Whenever two posets are order isomorphic, they can be considered to be "essentially the same" in the sense that one of...
between two well-ordered sets they have the same ordinal number. The first ordinal number that is not a natural number is expressed as ; this is also the ordinal number of the set of natural numbers itself.
Many well-ordered sets with cardinal number have an ordinal number greater than ω (the latter is the lowest possible). The least ordinal of cardinality (i.e., the initial ordinalThe von Neumann cardinal assignment is a cardinal assignment which uses ordinal numbers. For a well-ordered set U, we define its cardinal number to be the smallest ordinal number equinumerous to U. More precisely:...
) is .
For finite well-ordered sets, there is one-to-one correspondence between ordinal and cardinal numbers; therefore they can both be expressed by the same natural number, the number of elements of the set. This number can also be used to describe the position of an element in a larger finite, or an infinite, sequenceIn mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
.
Hypernatural numbers are part of a non-standard model of arithmetic due to Skolem.
Other generalizations are discussed in the article on numberA number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....
s.
Formal definitions
{{Main|Set-theoretic definition of natural numbers}}
Historically, the precise mathematical definition of the natural numbers developed with some difficulty. The Peano axioms state conditions that any successful definition must satisfy. Certain constructions show that, given set theorySet theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
, modelsIn mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
of the Peano postulates must exist.
Peano axioms
{{Main|Peano axioms}}
The Peano axiomsIn mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are a set of axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano...
give a formal theory of the natural numbers. The axioms are:
- There is a natural number 0.
- Every natural number a has a natural number successor, denoted by S(a). Intuitively, S(a) is a+1.
- There is no natural number whose successor is 0.
- S is injective, i.e. distinct natural numbers have distinct successors: if a ≠ b, then S(a) ≠ S(b).
- If a property is possessed by 0 and also by the successor of every natural number which possesses it, then it is possessed by all natural numbers. (This postulate ensures that the proof technique of mathematical induction
Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers...
is valid.)
It should be noted that the "0" in the above definition need not correspond to what we normally consider to be the number zero. "0" simply means some object that when combined with an appropriate successor function, satisfies the Peano axioms. All systems that satisfy these axioms are isomorphic, the name "0" is used here for the first element (the term "zeroth element" has been suggested to leave "first element" to "1", "second element" to "2", etc.), which is the only element that is not a successor. For example, the natural numbers starting with one also satisfy the axioms, if the symbol 0 is interpreted as the natural number 1, the symbol S(0) as the number 2, etc. In fact, in Peano's original formulation, the first natural number was 1.
A standard construction
A standard construction in set theorySet theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
, a special case of the von Neumann ordinal construction, is to define the natural numbers as follows:
- We set 0 := { }, the empty set
In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...
,
- and define S(a) = a ∪ {a} for every set a. S(a) is the successor of a, and S is called the successor function.
- By the axiom of infinity
In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of infinity is one of the axioms of Zermelo-Fraenkel set theory...
, the set of all natural numbers exists and is the intersection of all sets containing 0 which are closed under this successor function. This then satisfies the Peano axiomsIn mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are a set of axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano...
.
- Each natural number is then equal to the set of all natural numbers less than it, so that
- 0 = { }
- 1 = {0} = {{ }}
- 2 = {0, 1} = {0, {0}} = { { }, {{ }} }
- 3 = {0, 1, 2} = {0, {0}, {0, {0}}} = { { }, {{ }}, {{ }, {{ }}} }
- n = {0, 1, 2, ..., n-2, n-1} = {0, 1, 2, ..., n-2,} ∪ {n-1} = {n-1} ∪ (n-1) = S(n-1)
- and so on. When a natural number is used as a set, this is typically what is meant. Under this definition, there are exactly n elements (in the naïve sense) in the set n and n ≤ m (in the naïve sense) if and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
n is a subsetIn mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
of m.
- Also, with this definition, different possible interpretations of notations like Rn (n-tuples versus mappings of n into R) coincide.
- Even if the axiom of infinity fails and the set of all natural numbers does not exist, it is possible to define what it means to be one of these sets. A set n is a natural number means that it is either 0 (empty) or a successor, and each of its elements is either 0 or the successor of another of its elements.
Other constructions
Although the standard construction is useful, it is not the only possible construction. For example:
- one could define 0 = { }
- and S(a) = {a},
- producing
- 0 = { }
- 1 = {0} = {{ }}
- 2 = {1} ={{{ }}}, etc.
- Each natural number is then equal to the set of the natural number preceding it.
Or we could even define 0 = {{ }}
- and S(a) = a ∪ {a}
- producing
- 0 = {{ }}
- 1 = {{ }, 0} = {{ }, {{ }}}
- 2 = {{ }, 0, 1}, etc.
The oldest and most "classical" set-theoretic definition of the natural numbers is the definition commonly ascribed to Frege and RussellBertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, and social critic. At various points in his life he considered himself a liberal, a socialist, and a pacifist, but he also admitted that he had never been any of these things...
under which each concrete natural number n is defined as the set of all sets with n elements. This may appear circular, but can be made rigorous with care. Define 0 as {{ }} (clearly the set of all sets with 0 elements) and define S(A) (for any set A) as {x ∪ {y} | x ∈ A ∧ y ∉ x } (see set-builder notationIn set theory and its applications to logic, mathematics, and computer science, set-builder notation is a mathematical notation for describing a set by stating the properties that its members must satisfy...
). Then 0 will be the set of all sets with 0 elements, 1 = S(0) will be the set of all sets with 1 element, 2 = S(1) will be the set of all sets with 2 elements, and so forth. The set of all natural numbers can be defined as the intersection of all sets containing 0 as an element and closed under S (that is, if the set contains an element n, it also contains S(n)). One could also define "finite" independently of the notion of "natural number", and then define natural numbers as equivalence classes of finite sets under the equivalence relation of equipollence. This definition does not work in the usual systems of axiomatic set theory because the collections involved are too large (it will not work in any set theory with the axiom of separation); but it does work in New FoundationsIn mathematical logic, New Foundations is an axiomatic set theory, conceived by Willard Van Orman Quine as a simplification of the theory of types of Principia Mathematica. Quine first proposed NF in a 1937 article titled "New Foundations for Mathematical Logic"; hence the name...
(and in related systems known to be relatively consistent) and in some systems of type theoryIn mathematics, logic and computer science, type theory is any of several formal systems that can serve as alternatives to naive set theory, or the study of such formalisms in general...
.
See also
{{Col-begin}}
{{Col-1-of-3}}
- Canonical representation of a positive integer
- Countable set
In mathematics, a countable set is a set with the same cardinality as some subset of the set of natural numbers. A set that is not countable is called uncountable. The term was originated by Georg Cantor...
- Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
{{Col-2-of-3}}
{{Col-3-of-3}}
{{Portal|Mathematics}}
{{Col-end}}
External links
- Axioms and Construction of Natural Numbers
- Essays on the Theory of Numbers by Richard Dedekind
Julius Wilhelm Richard Dedekind was a German mathematician who did important work in abstract algebra , algebraic number theory and the foundations of the real numbers.-Life:...
at Project GutenbergProject Gutenberg is a volunteer effort to digitize and archive cultural works, to "encourage the creation and distribution of eBooks". Founded in 1971 by Michael S. Hart, it is the oldest digital library. Most of the items in its collection are the full texts of public domain books...
{{Number Systems}}
{{DEFAULTSORT:Natural Number}}